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Standing Waves
Standing Wave: A wave/vibration that has some particular points that remain in 'fixed' positions (nodes) and some points that vibrate with the maximum amplitude (antinodes). It is the superposition of two progressive waves moving in opposite directions. Key Information: * A standing wave can also be called a stationary wave. * A standing wave is not the same as a progressive wave, it transfers no energy. * When trying to simulate a standing wave using a vibration transducer, signal generator, and a string, most frequencies result in no pattern so a standing wave is not formed. * If the frequency changes to produce an exact number of waves in the time it takes for the waves to get to the edge of the string and back again, the waves will reinforce each other and produce a standing wave. * These frequencies are known as resonant frequencies. Formation: # Two (or more) waves travelling in opposite directions. # The waves must have equal frequencies and wavelengths. # The waves superpose. # A series of nodes and antinodes form. Standing Wave on a String: # Nodes form where the wave amplitude is zero. # Antinodes form where wave amplitude is at its maximum. # Resonant frequencies form when an equal number of half wavelengths fit onto the string. # At a standing waves lowest possible resonant frequency, the wave is at its fundamental frequency (first harmonic), the next lowest frequency is called the first overtone (second harmonic), the next is called the second overtone (third harmonic) etc. # On a string, the general equation for a standing wave is: L=(nX)/2 ''where L(length of vibrating string - m), n(number of half wavelengths) and X(wavelength - m). '''Standing Wave in a Pipe Closed at One End: # Standing waves can form in pipes with one end that is open, however, they show a slightly different pattern. # Instead of their being a node at both ends of the standing wave (like on a string, where reflection of the wave energy occurs), a node forms at the closed end of the pipe and an antinode at the open end of the pipe. # The general formula for standing waves in a pipe closed at one end is: '''''L+((2n+1)X)/4 where L(length - m), n(number of half wavelengths) and X(wavelength - m). # As shown in the image, the fundamental frequency (first harmonic) occurs when n=1. The second wave (where n=3) shows the first overtone (second harmonic) etc. Factors Affecting Resonant Frequencies: # The two factors that change the frequency at which a standing wave on a string will be formed are mass per unit length and tension. # With these values, the fundamental frequency and 'velocity' of a standing wave on string can be caculated. # The necessary equations are: -''T=mg ''where T(tension - N), m(mass - kg) and g(gravitational field strengh - 9.81N/kg). -''u=m/L'' where u(mass per unit length), m(mass - kg) and L(length of vibrating string - m). Hence, to calculate the fundamental frequency (and 'velocity') of the standing wave using these values, the following equations are needed: